LIBRARY OF 



CONGRESST] 



Shelf 



UNITED STATES OF AMERICA. 



OSBORN'S TABLES 



Moments of Inertia 



Squares of Radii of Gyration, 



FRANK C. OSBORN, C. E. 

r 



rc*> 



Pittsburgh, Pa. 






PUBLISHED BY 

ENGINEERING ERA PUBLISHING COMPANY 

CLEVELAND, OHIO. 



,0g 



Entered according to Act of Congress, 

in the year 1886, by 

FRANK C. OSBORN, C. E., 

a the office of the Librarian of Congres 

at Washington, D. C. 




CLARK-BRITTON PRINTING CO. PRINTERS, 
CLEVELAND, OHIO 



PREFACE. 

The aim of the author of the present work has been to supply, 
partially at least, a need felt by engineers and draughtsmen en- 
gaged in structural work in iron and steel. 

The handbooks of our manufacturers are all quite complete as 
far as the properties of their particular shapes are concerned, but 
there is believed to be no work published, containing a collection 
of built sections of the forms in general use at the present time. 

All of the sections given, are what are known as "open sec- 
tions," being accessible for examination and painting : a condition 
required by the majority of specifications. 

There will be found a number of sections which do not comply 
with the quite common specification that " the unsupported width 
of any plate subjected to compression, shall not exceed thirty 
times its thickness," the designer to use his discretion as to when 
these sections are admissible. 

It is to be noticed that the sections in each group are so pro- 
portioned that their centres of gravity are found at the same dis- 
tance above the centre of the web. 

There are no interpolations in any of the tables, each value 
having been calculated and checked independently, and the fig- 
ures given, may be relied upon as accurate. 

Pittsburgh, Feb., 1886. F. C O. 



EXPLANATION. 

The angle shapes used in the following tables are those manu- 
factured by the firm of Carnegie Bros. & Co., Limited, Pittsburg, 
Pa. The shapes of other manufacturers however, for the same 
size and weight of angle will not materially affect the values given 
for moment of inertia or radius of gyration. 

Dimensions are given in inches and weights in pounds per foot. 
In all cases where angles with unequal legs are used, the shorter 
leg is placed against the web. 

To illustrate the method of calculation used, examples from 
the tables will be taken. 

Table I, No. 1. To obtain moment of inertia and square of 
radius of gyration for an axis through center of gravity per- 
pendicular to web 



2 plates 12 X} = 6.00 sq. 
4 angles 2£x3 4.4 lbs. = 5.28 



H-12X12 2 = 72.00 

X5.34 2 =150.56 

4x0.74 = 2.96 



Total 11.28 sq. ir 

225.52=1 
225.52-K11.28=19.99=r 2 
The moment of inertia of the plates being *, bd 3 = ^ Ad 2 
in which b=breadth, d=the depth, and A the area of the plates, 
and the moment of inertia of each angle being ad 2 4- i, when a= 
the area of the angle, d the distance of its center of gravity from 
the neutral axis and i its moment of inertia about an axis through 
its own center of gravity perpendicular to the web. In the above 
example, 5.34 inches is the distance from center of gravity of angle 
to center of web, and 0.74 is the moment of inertia of one angle, 
about an axis through its center of gravity as given in Carnegie's 



Pocket Companion. The area of each angle is found from the 
weight per foot, by multiplying the latter by .3. 

Table II, No. 1. To obtain moment of inertia and square of 
radius of gyration for an axis through center of gravity per- 
pendicular to web. 

It will be convenient to first determine the position of the center 
of gravity, which is done as follows: Multiply the area of the top 
plate, top angles, webs and bottom angles, each by the distance of 
its center of gravity from the lower edge of web. Divide the sum 
of these products by the total area of the section, and the result 
will be the distance of the center of gravity above the lower edge 
of web. 

sq. in. 

Top plate 15x% 5.62x14.19=79.75 5.62 x 7. 19 2 =290.53 

2 top angles 3X3 5.9 lbs. 3.54x13.14=46.52 3.54X6.14 2 =133,46 

2 web plates U*% 10.50 x 7.00= 73.50 10.50x142+12 =171.50 

2 bot. angles 3x3 10.1 lbs. 6.06x0.98= 5.94 6.06X6.02 2 =219.62 

+ 8.51 

25.72x 8.00=205.71 

7.00 823.62 



deduct 25.72 



1=797.90 



r2 =797.90-25.72=31.02 

Find the moment of inertia of the section about an axis through 
the center of the web as follows: Multiply the area of the top 
plate, top angles and bottom angles each by the square of the 
distance of its center of gravity from the center of the web; add 
to these results the moment of inertia of the webs (^ Ad 2 ', and 
the moment of inertia of each angle about an axis through its 
center of gravity. From this result subtract the product of the 
area of the section by the square of the distance from the neutral 
axis to the center of the web, and the result will be the required 
moment of inertia of the section about an axis through the center 
of gravity perpendicular to the web. 

The moment of inertia of the top plate about an axis through 
its center of gravity should, strictly speaking, be added to the above; 
but its value is so small, that the final result is not materially 
affected. 

Tables III and IV differ from table II otily in the addition 
of flange plates, and the examples will fully explain the oper- 
ations performed. 



TABLE III. 

Top plate 21*% 10.50x18.25=191.58 

2 top ang. 3J^x3^ 8.0 lbs. 4.80x17.00= 81.60 

2 web plates 18x% 18.00x 9.00=162.00 

2 bot. ang. 3K X 4 11.6 lbs. 6.96X1.02= 7.10 

2 flan, plates 'SY^-Vt. 3.50x 1.75= 6.13 

43.76x10.25=448.41 
9.00 



10.50x9.25 2 =898.41 
4.80X8.00 2 =307.20 
18.00*12xl8 2 =486.00 
6.96X7.98 2 =436.26 
3.50X7.25 2 =183.97 
+ 17.15 



deduct 43.76xl.25' 

r 2 =2260.61*43.76=51.66. 



1=2260.61 



TABLE IV. 

Top plate 24x^ 12.00x20.25=243.00 

2 top ang. 3^x314 8.0 lbs 4.80x19.00= 91. 

2 web plates 20x^ 20.00x10.00=200.00 

2 bot. ang. 3^x4, 12.2 lbs. 7.32X 1.03= 7.54 

541.74 
2 flange plates 5x% 5.00x 0.25= 1.25 

49.12X11.00=540.49 
10.00 

deduct 49-12X LOO 2 



12.00xl0.25 2 =1260.75 

4.80X 9.00 2 = 388.80 

20.00*12X20 2 = 666.67 

7.32X 8.97 2 = 589.00 



5.00x10.25^ = 525.31 
+ 14.04 



.12 



1=3395.45 
r 2 =3395.45*49.12=69.13. 

The explanation given for table II will suffice for table V. 
the difference between them being only in the number of angles 
and plates. 

The word eccentricity is used in table II, III, IV, and V, to 
denote the distance of the center of gravity of the section above the 
center of the web. 

In the expression T ~ B which may be called a co-efficient of eccen- 
tricity, T represents the area of the top plate and top angles, B the 
area of the bottom angles and flange plates (if any are used) and 
A the total area of the section. This co-efficient is used in pro- 
portioning/or the same eccentricity, sections similar to those in the 
tables, but of different areas. 

Example. Kequired a section of 53.30 square inches, with an 
eccentricity of 1£ inches. 



8 

Referring to table II, we find 53.03 the area nearest to that 
required, and its co-efficient, .1469; now if T ~ B =.1469 

we will have T— B = A X.1469 =53.30X .1469= 7.83 

and T-f- B= A — area of webs =53.30— 27.00=26.30 

Hence by addition 2T =34.13 

and T =17.06 

B=26.30— 17.06 = 9.24 

17.06 — 13.13=3.93=area required in top angles 
=6.6 pounds per foot for each angle 

9.24-4- .6=15.4 " " " " " 

Proof. 

Top plate 21 Xf 13.13x18.31=240.41 

2 top angles 3x3 6.6 lbs. 3.96x17.12= 67.80 

2 web plates 18 Xf 27.00 X '.9.00=243.00 

2 bottom angles 3x4, 15.4 lbs. 9.24 X 0.95= 8.78 



53.33X10.50=559.99 
9.00 

1.50 
Example 2. Required a section of 44.84 square inches with an 
eccentricity of 1^ inches. 

Referring to table III, we find .1127 for the co-efficient corre- 
sponding to the area nearest to that required 

then T— B=44.84x .1127= 5.05 
and T-f B=44.84— 18.00=26.84 

2 T =31.89 

T=15.94 
B=26.84— 15.94 =10.90 

(15.94— 10.50)-t- .6=9.1 pounds per foot for top angle 
(10.90— 3.50)-r- .6=12.3 " " " " bottom angle. 
In the calculation for moment of inertia sideways, the distance 
out to out of webs was assumed equal to the width of top plates, less 
twice the nominal length of leg of top angle. 



TABLE NO. 1. 



I_ J 


■ 


r 

L 



Two Plates and Four Angles. 



TWO PLATES A^T'E 8 POUR ANGLES. 





Two 


FOUR ANGLES. 




AXI SAB. 


AXI S C D. 




Web Plates 
Size 




Total Area 
Square In. 












Size 


Lbs. per 










Out to Out 
Webs. 






in Inches. 


in Inches. 


Foot. 




X 


X- 2 


I 


1 


12 Xi- 


2£X3 


4.4 


11.28 


226 


19.99 


8.1 


227 


2 






5.0 


12.00 


245 


20.44 






3 






5.8 


12.96 


272 


20.95 






4 






6.7 


14.04 


300 


21.37 






5 






7.5 


15.00 


325 


21.70 






6 






8.3 


15.96 


349 


21.89 






7 






9.0 


16.80 


371 


22.06 


8.1 


383 


8 


12XA 


2£X3 


4.4 


12.78 


244 


19.05 


8.0 


243 


9 






4.6 


13.02 


250 


19.20 






10 






5.4 


13.98 


276 


19.77 






11 






6.2 


14.94 


303 


20.25 






12 






7.1 


16.02 


331 


20.65 






13 






7.9 


16.98 


355 


20.91 






14 






8.7 


17.94 


380 


21.16 






15 






9.0 


18.30 


389 


21.23 


8.0 


396 


16 


12 x| 


2JX3 


4.4 


14.28 


262 


18.31 


8.0 


263 


17 






5.0 


15.00 


261 


18.75 






18 






5.8 


15.96 


308 


19.27 






19 






6.7 


17.04 


336 


19.72 






20 






7.5 


18.00 


361 


20.08 






21 






8.3 


18.96 


385 


20.33 






22 






9.0 


19.80 


407 


20.53 


8.0 


416 


23 


14xf 


3X3 


5.9 


17.58 


444 


25.3 


9.5 


447 


24 






6.0 


17.70 


449 


25.4 






25 






6.5 


18.30 


470 


25.7 






26 






7.0 


18.90 


492 


26.1 






27 






8.0 


20.10 


535 


26.6 






28 






9.0 


21.30 


576 


27.1 






29 






10.0 


22.50 


617 


27.4 






30 






11.0 


23.70 


657 


27.7 


9.5 


669 


31 


14 X^ 


3X3 


5.9 


19.33 


473 


24.5 


9.5 


480 


32 






6.0 


19.45 


478 


24.6 






33 






6.5 


20.05 


499 


24.9 






34 






7.0 


20.65 


521 


25.2 






35 






8.0 


21.85 


563 


25.8 






36 






9.0 


23.05 


605 


26.2 






37 






10.0 


24.25 


646 


26.6 






38 






11.0 


25.45 


686 


26.9 


9.5 


701 



TWO PLATES a-'I 4 I' s FOUR ANGLES. 



XJE. 



Two 


FOLK ANGLES. 




AX I SAB. 


AXI S C D. 


W.h 1'latpB 




Total Area 
Square In. 












Size, 
in Inches. 








Size, 

in Inches - 


Lbg.per 

Fuut. 


I 


1*2 


W.bs. 


1 


39 


14X* 


3X3 


5.9 


21.08 


502 


23.8 


9.5 


512 


40 






6.0 


21.20 


506 


23.9 






41 






6.5 


21.80 


628 


24.2 






42 






7.0 


22.40 


550 


24.5 






43 






8.0 


23.60 


592 


25.1 






44 






9.0 


24.80 


o;:4 


25.6 






45 






10.0 


26.00 


674 


25.9 






46 






11.0 


27.20 


714 


26.3 


9.5 


733 


47 


15X1 


3X3 


5.9 


18.33 


529 


28.9 


10.2 


529 


48 






6.0 


18.45 


535 


29.0 






49 






6.5 


19.05 


560 


29.4 






50 






7.0 


19.65 


585 


29.8 






51 






8.0 


20.85 


635 


30.5 






52 






9.0 


22.05 


684 


31.0 






53 






10.0 


23.25 


732 


31.5 






54 






11.0 


24.45 


779 


31.9 


10.2 


776 


55 


15 X , 7 6 


3X3 


5.9 


20.20 


564 


27.9 


10.2 


570 


56 






6.0 


20.32 


570 


28.0 






57 






6.5 


20.92 


595 


28.4 






58 






7.0 


21.52 


620 


28.8 






59 






8.0 


22.72 


670 


29.5 






60 






9.0 


23.92 


719 


30.1 






61 






10.0 


25.12 


767 


30.5 






62 






11.0 


26.32 


814 


30.9 


10.2 


817 


63 


15X* 


3X3 


5.9 


22.08 


599 


27.1 


10.2 


mo 


64 






6.0 


22.20 


605 


27.2 






65 






6.5 


22.80 


630 


27.6 






66 






7.0 


23.40 


656 


28.0 






67 






8.0 


24.60 


705 


28.7 






68 






9.0 


25.80 


754 


29.2 






69 






10.0 


27.00 


802 


29.7 






70 




V 


11.0 


28.20 


849 


30.1 


10.2 


858 


71 


16X^6 


3X3 


5.9 


21.08 


666 


31.6 


10.9 


671 


72 






6.7 


22.04 


718 


32.4 






73 






7.5 


23.00 


759 


33.0 






74 






8.3 


23.96 


805 


33.6 






75 






9.2 


25.04 


856 


34.2 






76 






10.0 


26.00 


901 


34.6 






77 






10.8 


26.96 


945 


35.1 






78 






11.7 


28.04 


994 


35.4 


10.9 


985 



t^-zbue ?. 



TWO PLATES 



$£ 



b FOUR ANGLES. 





Two 


FOUR ANGLES. 


AXI SAB. 


AXI S C D. 




Web Plates 




; Total Area 










Size, 
in Inches, 


Size, i libs, per 
in Inches. | Foot. 


Square In. 


I 


r 2 


Out to Out 
Webs. 


X 


79 


16X* 


3X3 


100 


28.00 


943 


33.7 


10.9 


940 


80 






10.8 


28.96 


988 


34.1 






81 


16 XA 




10.0 


30.00 


986 


32.9 






82 






10.8 


30.96 


1031 


33.3 






83 


16X| 




10.0 


32.00 


1029 


32.2 






84 






10.8 


32.96 


1073 


32.6 






85 


16XH 




10.0 


34.00 


1071 


31.5 






86 






10.8 


34.96 


1116 


31.9 


10.9 


1125 


87 


18X* 


3X3 


5.9 


25.08 


961 


38.3 


12.2 


965 


88 






6.7 


26.04 


1023 


39.3 






89 






7.5 


27.00 


1083 


40.1 






90 






8.3 


27.96 


1144 


40.9 






91 






9.2 


29.04 


1230 


41.7 






92 






10.0 


30.00 


1269 


42.3 






93 






10.8 


30.96 


1327 


42.9 






94 






11.7 


32.04 


1391 


43.4 


12.4 


1384 


95 


18X| 


3x3 


6.3 


30.06 


1114 


37.0 


12.1 


1110 


96 






7.1 


31.02 


1175 


37.9 






97 






7.9 


31.98 


1235 


38.6 






98 




3JX3J 


8.8 


33.06 


1292 


39.1 






99 
100 
101 






9.6 
10.4 
11.3 


34.02 
34.98 
36.06 


1351 

1409 
1474 


39.7 
40.3 
40.9 






102 






12.1 


37.02 


1531 


41.4 


12.3 


1548 


103 
104 
105 
106 
107 
108 
109 


18xf 


3*X3£ 


8.3 
9.2 
10.0 
10.8 
11.7 
12.5 
13.3 


36.96 
38.04 
39.00 
39.96 
41.04 
42.00 
42.96 


1376 
1443 
1502 
1559 
1624 
1681 
1737 


37.2 
37.9 
38.5 
39.0 
39.6 
40.0 
40.4 


12.1 


1377 


110 






14.2 


44.04 


1800 


40.9 


12.3 


1827 


111 
112 
113 
114 
115 


21X| 


3*X3£ 


8.3 

9.0 

9.8 

10.6 

11.5 


36.21 
37.05 
38.01 
38.97 
40.05 


1873 
1946 
2029 
2113 
2203 


51.7 
52.5 
53.4 
54.2 
55.0 


14.2 


1877 


116 
117 






12.3 
13.1 


41.01 
41.97 


2284 
2367 


55.7 
56.4 






118 






14.0 


43.05 


2454 


57.0 


14.5 


2470 



c 
TWO PLATES a3 f |[ B FOUR ANGLES. 





Two 


FOUR ANGLES. 


AXIS A B. 


AXIS C D. 




Web Plates 

Size, 
in Inches. 




Total Area 
Square In. 










Size, 

in Inches. 


Lbs. per 
Foot. 


X 


r-2 


Out t..()ut 
Weba. 


X 


119 


21X| 


3JX3J 


8.8 


42.06 


2119 


50.4 


14.2 


2133 


120 






9.6 


43.02 


2202 


51.2 






121 






10.4 


43.98 


2285 


51.9 






122 






11.3 


45.06 


2377 


52.8 






128 






12.1 


46.02 


2458 


53.4 






124 






12.9 


46.98 


2539 


54.0 






125 






13.8 


48.06 


2629 


54.7 






126 






14.6 


49.02 


2707 


55.2 


14.4 


2713 



TABLE NO. 2. 



~l 



_J 

Three Plates and Four Angles 



r 



3c 


H 


CO CO-* 

OS 3 CO 
lO CO CO 


t~ rH rH 

HtOH 

rooso 


ic co co ■># oo io ic 

OH05NU5<SM 
CO CO CO t~ t~ t~ 00 


CO-* O CO rH 
rHrH COCO CO 


HI 


iOtF CO 

3 rH rH 


<M <M -c)H 

(MINN 


00 CO OO lO 00 CO OS 
OS rfH 00 a> CO lO ■* 

r-l >-( l-H rH CO CO CO 
rH rH rH rH l— I i— 1 I— I 


loaoHO 


w S 


ooo 

3 3 3 


OOJO 
HOH 


lOHfflfflOHO 

qqaaqqq 

HrlddrHHrt 


HHOJOO 

ooaao 
hh'oo'h 


cq 


U 


OHN 
l-ii-irH 
CO CO CO 


ON<M 
NNN 

CO CO CO 


rjj lO CO ^j >o lO CO 

id id co cd co co cd 

eo eo co co co co co 


q q « n co 

CO CO CO CO CO 


H 


00O3W 

OS 3 ■* 
t-00 00 


OH1IJ 

CO 3 CO 
00 OS OS 


rr< CO rH ^H CO CO tJH 
CO "* CO lO OS CO C-- 

OlOOOOHH 


U5NU5UMO 

lONHlOO) 

33 rH r-l rH 






Total Area 
Square In. 


NNQO 

i>qo5 

id CO CD 


(MC0O 

qc»q 

I>!>05 
N(M(M 


CO CD CO O CD CO CO 
«* CD rH O 05 05 q 
CD CO 00 05 05 O CO 
(M (M CM (M (M C0 CO 


COCOON'* 
10 05000 

cd co d ■-< c4 

NfMSOCOCO 


W 
3 
< 

o 

H 
O 


i £ 


HMO 
O O* t-? 


rH H CO 






02 l-l 


CO 

X 

co 


X 

CO 


co ■* 
X X 

co co 


X 

co 


o 
< 

Ph 

o 


3 ^ 


OS CO rH 
lO CO t^ 


Ol 00 00 


os o io cq cq cq cq 

ui ® N CO OS d H 


as cq co co co 
odt>co'aJ 


sal 


CO 

X 

CO 


CO 

X 

eo 


CO 

X 

CO 


CO 

X 

CO 


Top Plate. 


M|00 

X 

lO 


co|co 

X 


ccloo 

X 

lO 


X 

00 


H .S 


co|oo 

X 


co|co 

X 


X 


«(Q0 

X 

rH 






rH CO CO 


^U5CO 


t- 00 OS © rH CO CO 


t* LO CO t^OO 



09 


i H 


(MO 01i60ieNN-»<HMO)3)OHTj(sON 

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Jl IL 



Five Plates and Four Angles. 





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TABLE NO. 4. 



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TABLE NO. 5. 




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Four Plates and Eight Angles. 



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MEMORANDUM. 



MEMORANDUM. 



MEMORANDUM. 



